e3x.nn.functions.chebyshev.exponential_chebyshev

e3x.nn.functions.chebyshev.exponential_chebyshev(x, num, gamma=1.0, cuspless=False, use_exponential_weighting=False)

Exponential Chebyshev polynomial basis functions.

Computes the basis functions (see basic_chebyshev and exponential_mapping)

\[\mathrm{exponential\_chebyshev}_k(x) = \mathrm{chebyshev}_k(2\cdot\mathrm{exponential\_mapping}(x)-1)\]

or (if use_exponential_weighting = True)

\[\mathrm{exponential\_chebyshev}_k(x) = \mathrm{exponential\_mapping}(x) \cdot \mathrm{chebyshev}_k(2\cdot\mathrm{exponential\_mapping}(x)-1)\]

where \(k=0 \dots K-1\) with \(K\) = num.

Plot for \(K = 5\) and \(\gamma = 1\) (cuspless = False, use_exponential_weighting = False):

Plot for \(K = 5\) and \(\gamma = 1\) (cuspless = False, use_exponential_weighting = True):

Plot for \(K = 5\) and \(\gamma = 1\) (cuspless = True, use_exponential_weighting = False):

Plot for \(K = 5\) and \(\gamma = 1\) (cuspless = True, use_exponential_weighting = True):

Parameters:

• x (<class 'Float[Array, '...']'>) – Input array.

• num (int) – Number of basis functions \(K\).

• gamma (Union[Float[Array, ''], float], default: 1.0) – Exponential decay constant for the exponential mapping.

• cuspless (bool, default: False) – If True, the returned functions are cuspless.

• use_exponential_weighting (bool, default: False) – If True, the functions are weighted by the value of the exponential mapping.

Return type:

<class 'Float[Array, '... num']'>

Returns:

Value of all basis functions for all values in x. The output shape follows the input, with an additional dimension of size num appended.